We consider the lossless Kaspi/Heegard-Berger source coding problem where an encoder communicates a common description of two sources to two decoders, and each decoder wants to reconstruct one of the sources with the help of side information. We present new results on the utility of encoder side information for this scenario. We show that for some sources and side informations---e.g., for some instances of conditionally less noisy side information---the minimum rate that is required to describe the sources is strictly reduced when the side information is also known at the encoder. On the other hand, we identify classes of sources and side informations---e.g., physically degraded side information---where encoder side information does not change the minimum description rate. %evenknowledge of the side information at the encoder does not reduce the minimum description rate. But we also demonstrate that for some sources and side informations outside these classes---e.g., for some instances of conditionally less noisy side informations---encoder side information reduces the minimum rate.

We show similar results for a scenario where one decoder has to reconstruct both sources and for a scenario where the encoder is informed only about one of the decoder's side information.